Analysis of Historic StructuresAnalysis of Historic Structures
Lecture 2: Intro to Masonry Structure
Review of Last MeetingReview of Last Meeting
Principles of historic structure– EQUILIBRIUM
Lower bound and upper bound theorems
Possible research topics
Lower Bound TheoremLower Bound Theorem
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Upper Bound TheoremUpper Bound Theorem
λP
Theorems of Limit AnalysisTheorems of Limit Analysis
1. Lower Bound: If you can demonstrate at least one possible equilibrium state, then the structure can also find at least one possible stable state
2. Upper Bound: When the load path can no longer be contained within the structure, and it is the unique and largest possible load, then it is the collapse load
Hooke’sHooke’s “2“2ndnd” Law (1675)” Law (1675)
“ut pendet continuum flexile, sic stabit
contiguum rigiduminversum”
As hangs the flexible line, so but inverted will stand the rigid
arch.
Compression vs. TensionCompression vs. Tension
Compression Tension
Equilibrium at a PointEquilibrium at a Point
ΣF = 0 (sum of forces is zero)
Compression
Tension
Structural EquationsStructural Equations
Only three types of equations:
1) Equilibrium2) Material properties (elasticity, etc)3) Compatibility (geometry)
We will focus on equilibrium equations because they are the most important.
Graphic Graphic StaticsStatics
Applet by Simon Greenwold
Hooke’sHooke’s Hanging ChainHanging Chain
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Design of MasonryDesign of Masonry
Main principle: must be kept in compression
Also applies to cast iron, unreinforced concrete, and other “brittle” materials
3D Vaults: “Slice” into arches3D Vaults: “Slice” into arches
Structural Analysis of MasonryStructural Analysis of Masonry
The Stone Skeletonby Jacques Heyman
Three main assumptions:– No tensile strength– Infinite compressive strength (rigid)– Sliding does not occur
Arch on Spreading SupportsArch on Spreading Supports
Range of Arch ThrustRange of Arch Thrust
Internal thrust lines due to self weight of arch
Range of Arch ThrustRange of Arch Thrust
TENSION
HmaxHmin
Range of Arch ThrustRange of Arch Thrust
COMPRESSION
Model Arch ExperimentModel Arch Experiment
Model Arch at Collapse StateModel Arch at Collapse State
Understanding cracks in masonryUnderstanding cracks in masonry
1. Why do cracks occur?
2. What do they tell us?
3. Are they a cause for concern?
Understanding cracks in masonryUnderstanding cracks in masonry
1. Why do cracks occur?-Small movements of supports
2. What do they tell us?-Where forces are NOT acting
3. Are they a cause for concern?-Usually not, but they can be
Understanding of CollapseUnderstanding of Collapse
Causes of collapse:
1. Displacements-Foundation movements, mortar “creep” over time
2. Overloading (truck on a bridge)-Water on vaults, collapsing roof on vault
3. Accelerations-Vibrations, earthquakes
Design and Analysis of Design and Analysis of UnreinforcedUnreinforced MasonryMasonry
Stability rather than failure of the material is the dominant concern
Collapse occurs when the load path can no longer be contained within the masonry
Model Arch at Collapse StateModel Arch at Collapse State
Single Span Stone ArchSingle Span Stone Arch
Single Span ArchSingle Span Arch
Single Span ArchSingle Span Arch
Single Span ArchSingle Span Arch
Double Span Stone ArchDouble Span Stone Arch
Double Span Arch with FillDouble Span Arch with Fill
Double Span Arch with FillDouble Span Arch with Fill
Double Span Arch with FillDouble Span Arch with Fill
Double Span Arch with FillDouble Span Arch with Fill
•This makes sense for bridges, but buildings don’t usually have trucks driving on top of the vaults
•Deformation over time can cause collapse in buildings
ConclusionsConclusions
Unreinforced masonry structures have very low stress levels: stability, not strength, governs the safety
Determine collapse states based on thrust line analysis using graphic statics
Equilibrium equations are most important when analyzing historical structures